Let's analyze the given expression [tex]\(\left(2^3\right)^{-5}\)[/tex] step-by-step and determine which of the options is equivalent to it.
1. Simplifying the Expression:
The given expression is [tex]\(\left(2^3\right)^{-5}\)[/tex]. To simplify this, we can use the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Here, [tex]\(a = 2\)[/tex], [tex]\(m = 3\)[/tex], and [tex]\(n = -5\)[/tex].
[tex]\[
\left(2^3\right)^{-5} = 2^{3 \cdot (-5)} = 2^{-15}
\][/tex]
2. Interpreting the Result:
We now have [tex]\(2^{-15}\)[/tex]. When dealing with a negative exponent, we recall that [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. So, [tex]\(2^{-15}\)[/tex] can be written as:
[tex]\[
2^{-15} = \frac{1}{2^{15}}
\][/tex]
Therefore, the simplified equivalent expression is [tex]\(\frac{1}{2^{15}}\)[/tex].
3. Comparison with Given Options:
Now we compare this result with the given multiple-choice options:
- [tex]\(\frac{1}{2^{15}}\)[/tex]
- [tex]\(\frac{1}{2^8}\)[/tex]
- [tex]\(2^8\)[/tex]
- [tex]\({ }_2 15\)[/tex]
By comparing, we see that the correct answer is [tex]\(\frac{1}{2^{15}}\)[/tex].
Thus, the expression equivalent to [tex]\(\left(2^3\right)^{-5}\)[/tex] is:
[tex]\[
\boxed{\frac{1}{2^{15}}}
\][/tex]
